WP 34S binomial bug?

[Dieter]

(05-03-2015 12:24 AM)rprosperi Wrote:  In the interim, perhaps someone that knows how these items are defined could provide them here for others to reference?

Pauli already noted the essential point, but let me add a more general explanation.

The cumulative distributions calculate the probability that the random variable is ≤ a given value (Norml, Binom, Poiss etc.) resp. that it's ≥ this value (Norml_u, Binom_u, Poiss_u etc.). The suffix u was chosen since the result mathematically is the upper tail integral (resp. the upper sum) of the probability density/mass function.

If the random variable is continuous (i.e. it can have real values like 3.7 or –1.638) the lower integral is bounded by –infinity and x, and the upper integral is bounded by x and +infinity. Both sum up to 1, so e.g. Norml_u(x) = 1 – Norml(x).

If the random variable is discrete its can take only integer values: a random experiment may have three or four successes, but not 3.74. Here the lower CDF (Binom, Poiss etc.) is the sum of the probabilities for 0, 1, 2, .... x successes, while the upper CDF (Binom_u, Poiss_u etc.) is the sum for x, x+1, x+2, ... n successes. You see that the probabilty for exactly x successes occurs in both sums, so they do not sum up to 1. However Binom(x) + Binom_u(x+1) = 1. In other words: the probabilty for "up to 4 successes" is 1 – the probability for "5 and more successes".

Re. Poiss and Poissλ: Yes, the former version (with two parameters whose product is λ again) actually is less common, and it could have been omitted. The one-parameter Poissλ is the usual way the Poisson distribution is defined. I learned the two-parameter version was included because this way it could be used with the same parameters (p and n) as the Binomial distribution. Since for large n and small p the Binomial approaches the Poisson distribution I do not see much sense in this, but I don't think this will get changed.

BTW Pauli: I am still working on the Poisson and Binomial quantile functions, and I think they work quite well now. At least they are more accurate and faster than the current code, and they work even in some border cases. Which does not mean they are perfect. ;-) The complete code for both cases now has 260 lines. This includes real results. I think I'll find some time later this month to take a closer look at this so that the 34s may get an update. In this case Walter will have to update the manual because there is a new feature that allows both integer and real results for the quantile (user selectable by setting a flag).

Dieter